Uncertainties in theoretical predictions for 						γ						d						→																					π																						0																			d					 observables near threshold due to the use of different elementary amplitudes

Al-Ghamdi, H. M.; Darwish, E. M.; Ibraheem, A. A.; Abou-Elsebaa, H. M.; Al-Ghamdi, H. M.; Darwish, E. M.; Ibraheem, A. A.; Abou-Elsebaa, H. M.

doi:10.31349/revmexfis.68.061201

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Revista mexicana de física

Print version ISSN 0035-001X

Rev. mex. fis. vol.68 n.6 México Nov./Dec. 2022 Epub July 31, 2023

https://doi.org/10.31349/revmexfis.68.061201

Research

Nuclear Physics

Uncertainties in theoretical predictions for γd→π0d observables near threshold due to the use of different elementary amplitudes

H. M. Al-Ghamdi^a

E. M. Darwish^b^c^*

A. A. Ibraheem^d^e

H. M. Abou-Elsebaa^b^c

^{^a} Physics Department, College of Science, Princess Nourah bint, Abdulrahman University, P. O. Box 84428, Riyadh 11671, Saudi Arabia.

^{^b} Physics Department, Faculty of Science, Taibah University, Medina 41411, Saudi Arabia.

^{^c} Physics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt.

^{^d} Physics Department, Faculty of Science, King Khalid University, Abha 62529, Saudi Arabia.

^{^e} Physics Department, Faculty of Science, Al-Azhar University, Assiut Branch, Assiut 71524, Egypt.

Abstract

We discuss possible uncertainties in theoretical predictions for γd→π0d observables near threshold due to the use of different elementary γN→πN amplitudes using an approach which is based on time-ordered perturbation theory. Results are presented for unpolarized cross sections and all possible spin asymmetries of differential and total cross sections. Our results indicate that the estimations of the uncertainty on the γd→π0d observables show important sensitivity to the modeling of the elementary γN→πN operator. A comparison to presently available experimental data is given. The results presented here are of particular interest for the evaluation of the systematic uncertainties caused by the use of different elementary operators in the analyses of γd→π0d measurements to extract information on the free neutron amplitude from deuteron data.

Keywords: Meson production; photoproduction reactions; few-body systems, deuteron; polarization phenomena in reactions; spin observables; polarized beams; polarized targets

1 Introduction

Understanding the internal structure and reactions of nuclear systems from first principles was, for a long time, an important goal of nuclear and particle physics. In this regard, few- and many-body systems provide a unique laboratory for studying nuclear interactions. Recent years have witnessed interesting developments in coherent and incoherent pion photoproduction on the deuteron and light nuclei, since relevant experimental studies have been performed (see Refs. ^[¹^-¹⁴^] and references therein). These reactions can be utilized to investigate the internal structure of hadrons in the non-perturbative domain of Quantum Chromodynamics (QCD) and study the behavior of nucleon resonances in the nuclear medium.

In particular, the reaction γd→π0d may be used as an isospin filter and it is sensitive to the coherent sum of the proton and neutron amplitudes γp→π0p and γn→π0n, respectively. The γd→π0d reaction near π-threshold serves also as a test of our understanding of the chiral πN dynamics. The use of deuteron as an effective neutron target allows one to gain abundant information on the mechanisms of the elementary pion photoproduction off free neutron which otherwise is not possible due to the absence of any stable, dense, free-neutron targets. Furthermore, the deuteron represents also an ideal object for the study of NN interactions.

So far, many studies have been carried out dealing with the reaction γd→π0d in the photon energy region from π-threshold up to 1 GeV in order to obtain information on π⁰-photoproduction off free neutrons ^[¹⁵^-³³^]. In spite of all these great theoretical efforts, a good description of the experimental data has not been yet found ^[³^,⁴^,⁶^,⁷^]. The disagreement between theoretical models and experimental data may be indicative of shortcomings in the elementary γN→πN amplitudes. It was found in Refs. ^[³⁴^-³⁹^] that the γd→πNN observables near 𝜋-threshold are considerably dependent on the elementary γN→πN amplitude. During the last decades, more realistic models for the γN→πN amplitude have been constructed ^[⁴⁰^-⁴⁸^]. Therefore, the construction of more realistic models for the investigation of the electromagnetic pion photo- and electroproduction processes on the deuteron and light nuclei remain challenging.

In addition, theoretical models which consider polarization observables for the reaction γd→π0d near π-threshold are very rare. Indeed, polarization observables constitute much more stringent tests of theoretical models. For instance, the helicity dependent total cross sections with respect to parallel (σ^P) and antiparallel (σ^A) spins of photon and deuteron provide important information on the nucleon spin structure. These cross sections are also required to test the validity of the Gerasimov-Drell-Hearn (GDH) sum rule ^[⁴⁹^]. Furthermore, the deuteron spin asymmetry σ^P — σ^P and the helicity E-asymmetry contain very interesting physics with respect to the internal hadron structure. Moreover, the double spin asymmetry T10c for circular polarized photons and vector polarized deuterons is also of great interest. It gives additional important information on the internal spin structure of the deuteron and provides a direct extraction of the GDH sum rule ^[⁴⁹^].

Therefore, the main aim of the present work is to discuss theoretical uncertainties in the analyses of γd→π0d observables near 𝜋-threshold due to the use of elementary γN→πN amplitudes. Since the extraction of the elementary γp→π0p and γn→π0n amplitudes essentially relies on an interpretation in terms of the plane-wave impulse approximation (PWIA), it is necessary to study its validity, i.e., to have a quantitative estimate of all possible mechanisms going beyond the PWIA. However, in the present paper we want to restrict ourselves to the PWIA while a more realistic treatment including all possible reaction mechanisms will be reported in a forthcoming paper. Here, we want to explore the impact on the description of the γd→π0d observables from the choice of the pion photoproduction amplitudes off protons and neutrons. To the best of our knowledge, this has not been studied previously in the literature. We also investigate whether the uncertainties caused by the use of different elementary operators for the predictions of polarization observables in the γd→πNN reaction channels ^[³⁴^-³⁹^] are also seen in the analyses of the coherent π⁰-photoproduction off deuterons, γd→π0d. It was demonstrated in Ref. ^[¹^] that the single- and double-polarization observables will allow us to select between different models for pion photoproduction on the nucleon.

Most recently, the sensitivity of the results for single-spin asymmetries and the helicity E-asymmetry in the γd→π0d reaction near threshold to the choice of the pion photoproduction amplitude has been investigated ^[³³^]. Unfortunately, the results presented in that work are based on an error in the definitions of initial and final relative momenta of the two nucleons in the deuteron process. Because of that the corresponding total energy of the photon-nucleon (γN) subsystem was not properly defined, but it strongly affects the predicted results especially at photon lab-energies close to π-threshold and forward pion angles. Thus, the importance of theoretical uncertainties in the γd→π0d observables still requires a more careful treatment as done in Ref. ^[³³^].

Therefore, the present work was motivated to present results for the unpolarized differential and total cross sections as well as for all possible beam, target, and beam-target spin asymmetries of the differential and total cross sections in the photon energy region from near π-threshold to 170 MeV. As elementary amplitudes, the one provided by the unitary isobar MAID-2007 model from ^[⁴⁶^] and those obtained using the Dubna-Mainz-Taipei dynamical model (DMT-2001) ^[⁴⁷^] and the chiral MAID model (χMAID-2013) ^[⁴⁸^] are used. Furthermore, we compare our results for the unpolarized differential cross section with the experimental data from TAPS ^[⁴^]. In this comparison with experiment we concentrate our discussion on the unpolarized differential cross section, because data for polarization observables are still not available. The calculations presented in this paper are of particular interest for the evaluation of the systematic uncertainties caused by the use of different elementary operators in the analyses of γd→π0d measurements. These uncertainties should be kept in mind when extracting information on the free neutron amplitude from experimental data on the deuteron.

This paper is structured as follows. In Sec. 2, we briefly outline the theoretical framework used to describe the γd→π0d reaction. We give the explicit expressions for the γd→π0d observables in terms of the reduced t-matrix elements. Subsequently, we calculate the scattering amplitude for this reaction based on time-ordered perturbation theory. Section 3 is devoted to the main results together with a comparison with the available experimental data. Finally, we summarize our results in Sec. 4.

2 Theoretical framework

In this section we briefly describe the theoretical framework for the coherent π⁰-photoproduction reaction on the deuteron. We would like to mention that the general formalism for the γd→π0d reaction has been described in Ref. ^[²⁹^], and we refer the reader to that work for further details. Here, we briefly recall the necessary notation and definitions.

2.1 Cross section and observables

We consider the reaction γd→π0d in the photon-deuteron (γd) center-of-momentum (c.m.). We use a coordinate system with the z-axis along the photon momentum k→e→z=k^=k→/k, the y-axis parallel to k→×q→, and the x-axis in the direction of maximal linear photon polarization ^[⁵⁰^], in which the pion momentum q→ has spherical coordinates θ and ϕ. If the incoming photon beam is not linearly polarized, then the x-axis may be chosen arbitrary, and there is no dependence on the angle ϕ. For the deuteron states, a non-covariant normalization following to the conventions of Ref. ^[⁵¹^] is used. The initial and final total three-momenta of the two nucleons in the deuteron are given in the c.m. system by P→=p→1+p→2=-k→ and P→'=p→ '1+p→ '2=-q→, respectively. For the initial and final relative three-momenta of the two nucleons in the deuteron in the c.m. system, we obtain p→r=p→1-p→2/2=p→-k→/2 and p→ 'r=p→ '1-p→ '2/2=p→-q→/2, respectively (see Fig. 1).

Figure 1 Diagrammatic representation of the γd→π0d reaction in the impulse approximation with definition of momenta in the γd c.m. system. The spectator nucleon is on-shell.

The cross section for arbitrary polarized photons and initial deuterons can be calculated for a given transition M-matrix by applying the density matrix formalism ^[⁵⁰^]. All different cross sections for the γd→π0d reaction have the form:

O=K∑m'~dλ~m~d‍∑m'dλmd‍Mm'~dλ~m~d*Ωm'~dm'dMm'dλmdρλλ~γ ρmdm~dd , (1)

where K is a kinematic factor, Mm'dλmd the scattering matrix, ρ^γ(ρ^d) the density matrix for incoming photon (initial deuteron) polarization, mdm'd the spin projection of the initial (final) deuteron, and λ ± 1 the circular photon polarization. A polarimeter for the deuteron in the final state is described by the operator Ω. In the present work, we set Ω = 1, because we do not consider any polarization analysis of the final deuteron. The scattering Mm'dλmd-matrix is given by isolating the azimuthal dependence as follows ^[¹⁷^,²²^]:

Mm'dλmd(θ,ϕ)=ei(λ+md)ϕ tm'dλmd(θ) , (2)

where the reduced t-matrix elements are defined by separating the ϕ-dependence from the M-matrix elements. Parity conservation gives for the t-matrix the symmetry:

t-m'd-λ-md=(-1)1+m'd+λ+md tm'dλmd . (3)

The kinematic factor K is given in the γd c.m.frame by:

K=116π2EdE'dWγd2 qk , (4)

where the deuteron energies in the initial and final states are given by Ed=Md2+k2 and E'd=Md2+q2, respectively, with M_d as deuteron mass. The absolute values of the photon and pion three-momenta in the γd c.m. frame are given, respectively, by:

k=12Wγd (Wγd2-Md2)andq=12Wγd Wγd2-(Md-mπ)2Wγd2-(Md+mπ)2 . (5)

The invariant energy of the γd system is given as:

Wγd=Eγ+Ed=k+Md2+k2 ,=Eπ+E'd=mπ2+q2+Md2+q2 ,=Md2+2MdEγ , (6)

Where m_π is the neutral-pion mass.

Following the rules of Ref. ^[⁵²^], we write the differential cross section in terms of the unpolarized differential cross section dσ₀/dΩ and the various spin asymmetries Σ, TIM, TIMc, and TIMl as follows:

dσdΩ=dσ0dΩ1+Plγ {Σ cos2ϕ+∑I=12‍PId ∑M=-II‍TIMlcosψM-δI1 π2 dM0I(θd)}+∑I=12‍PId ∑M=0I‍(TIMcosM(ϕd-ϕ)-δI1 π2+Pcγ TIMcsinM(ϕd-ϕ)+δI1 π2) dM0I(θd), (7)

where ψM=M (ϕd-ϕ)+2ϕ and dM0I(θd) is a small rotation matrix for which the convention of Ref. ^[⁵³^] is used. The photon polarization is characterized by the degree of circular Pcγ=Pzγ and linear Plγ=(Pxγ)2+(Pyγ)2 polarization, where the x-axis has been chosen in the direction of maximum linear polarization, i.e. Pxγ=-Plγ and Pyγ=0. The deuteron is characterized by the vector and tensor polarization components P1d and P2d, respectively, and the orientation angles (θd,ϕd) of the orientation axis of the deuteron with respect to which the density matrix of the deuteron has been assumed to be diagonal; the elements with md≠md' will therefore vanish.

In order to express the unpolarized differential cross section and various polarization observables in terms of the t-matrix, it is appropriate to define the two quantities ^[⁵⁰^]:

VIM=K3 2I+1 ∑mdmd'‍(-)1-md11Imd'-mdM∑md''‍tmd''1md'* tmd''1md , (8)

WIM=-K3 2I+1 ∑mdmd'‍(-)1-md11Imd'-mdM∑md''‍tmd''1md'* tmd''-1md , (9)

where the convention of Edmonds ^[⁵³^] is used for the Wigner 3j-symbol. The explicit formal expressions for unpolarized differential cross section and spin asymmetries can be derived as follows ^[²⁸^]:

(i) The unpolarized differential cross section:

dσ0dΩ=V00=K3 ∑m'dmd‍|tm'd1md|2 . (10)

(ii) The linear photon asymmetry:

Σ dσ0dΩ=W00=-K3 ∑m'dmd‍tm'd1md* tm'd-1md . (11)

(iii) The vector target asymmetry:

T11 dσ0dΩ=2 Im V11=23 K Im ∑md‍(tmd1-1* tmd10+tmd10* tmd11) . (12)

(iv) The tensor target asymmetries:

T2M dσ0dΩ=(2-δM0) Re V2Mfor0≤M≤2,

with

T20 dσ0dΩ=K32 ∑md‍(|tmd11|2+|tmd1-1|2-2 |tmd10|2) , (13)

T21 dσ0dΩ=23 K Re ∑md‍ (tmd1-1* tmd10-tmd10* tmd11) , (14)

T22 dσ0dΩ=2K3 Re ∑md‍ tmd1-1* tmd11 . (15)

(v) The spin asymmetries for circularly polarized photons and vector polarized deuterons:

T1Mc dσ0dΩ=-(2-δM0) Re V1M ,for0≤M≤1

with

T10c dσ0dΩ=K6 ∑md‍(|tmd11|2-|tmd1-1|2) , (16)

T11c dσ0dΩ=-23 K Re ∑md‍(tmd1-1* tmd10+tmd10* tmd11) . (17)

(vi) The spin asymmetries for circularly polarized photons and tensor polarized deuterons:

T2Mc dσ0dΩ=-2-δM0Im V2M ,for0≤M≤2

with

T21c dσ0dΩ=23 K Im ∑md‍(tmd10* tmd11-tmd1-1* tmd10) , (18)

T22c dσ0dΩ=-2K3 Im ∑md‍tmd1-1* tmd11 . (19)

Because the quantity VI0 is real according to VIM*=(-)M VI-M under complex conjugation, the spin asymmetry T20c vanishes identically.

(vii) The spin asymmetries for linearly polarized photons and vector polarized deuterons:

T1Ml dσ0dΩ=i W1M,for-1≤M≤1,

with

T10l dσ0dΩ=23 K Im ∑md‍tmd11* tmd-11 , (20)

T11l dσ0dΩ=-23 K Im ∑md‍tmd1-1* tmd-10, (21)

T1-1l dσ0dΩ=23 K Im ∑md‍tmd11* tmd-10 . (22)

(viii) The spin asymmetries for linearly polarized photons and tensor polarized deuterons:

T2Ml dσ0dΩ=W2Mfor-2≤M≤2,

with

T20l dσ0dΩ=23 K Re ∑md‍(tmd10* tmd-10-tmd11* tmd-11) , (23)

T21l dσ0dΩ=23 K Re ∑md‍tmd10* tmd-11 , (24)

T2-1l dσ0dΩ=23 K Re ∑md‍tmd10* tmd-1-1 , (25)

T22l dσ0dΩ=-K3 ∑md‍tmd1-1* tmd-11 , (26)

T2-2l dσ0dΩ=-K3 ∑md‍tmd11* tmd-1-1 . (27)

We would like to mention that for θ = 0 and π the spin asymmetries TIMc=0 for M≠0 and TIMl=0 for M≠2 because in that case the angle ϕ is undefined or arbitrary and, therefore, the differential cross section cannot depend on ϕ.

The deuteron spin asymmetry with respect to circularly polarized photons and the deuteron spin oriented parallel (P) and antiparallel (A) to the photon spin is related to the spin asymmetry T10c according to ^[⁵⁴^]:

d(σP-σA)dΩ=6 dσ0dΩ T10c . (28)

As mentioned in the Introduction, the asymmetry T10c is of special interest, because it is related to the spin asymmetry σ^P — σ^A which determines the GDH sum rule ^[⁴⁹^].

The general form of the total cross section with inclusion of photon and deuteron polarization effects is obtained from Eq. (7) by integrating dσ/dΩ over the pion spherical angle dΩ and reads ^[⁵⁴^]:

σ(Plγ,Pcγ,P1d,P2d)=σ0 1+P2d T~20 12(3cos2θd-1)+Pcγ P1d T~10 c cosθd+Plγ P2d T~22 lcos(2ϕd) 64sin2θd , (29)

where the unpolarized total cross section σ₀ and the corresponding spin asymmetries σ0 T~20, σ0 T~10c and σ0 T~22l are given by:

σ0=∫‍dΩ dσ0dΩ , (30)

σ0 T~20=∫‍dΩ dσ0dΩ T20 , (31)

σ0 T~10c=∫‍dΩ dσ0dΩ T10c , (32)

σ0 T~22l=∫‍dΩ dσ0dΩ T22l . (33)

2.2 The γd→π0d amplitude

Next, the transition matrix elements Mm'dλmd are calculated in the frame of time-ordered perturbation theory. The impulse approximation (IA), in which the reaction will take place only on one of the two nucleons in the deuteron leaving the other as a pure spectator, usually serves as the starting point to calculate the amplitude for electromagnetic pion production on the deuteron ^[¹⁶^] or, in general, on a nucleus. It corresponds to a direct embedding of the elementary γN→πN amplitudes into the two-nucleon system. In our framework, we restrict the calculation of the transition matrix elements Mm'dλmd to IA in order to study the dependence of results for the γd→π0d observables on the elementary pion photoproduction operator.

The production operator tγπd for the deuteron process is obtained from the elementary operator tγπ by:

tγπd=tγπ(1)⊗¶2+¶(1)⊗tγπ(2) . (34)

The upper index refers to the nucleon on which the elementary operator acts. This means that tγπd contains pure single nucleon terms. The transition M-matrix of coherent π⁰-photoproduction on the deuteron has the following form:

Mm'dλmd(k→,q→)=2∫‍d3p(2π)3 ψm'd†p→-q→2× 〈p→-q→|tγπ(1)|p→-k→〉 ψmdp→-k→2 , (35)

where ψmd(p→) denotes the deuteron wave function and tγπ(1) the elementary γN→πN operator.

For the intrinsic part of the deuteron wave function we use the ansatz:

ψmd(p→ )=∑L=0,2‍∑mLmS‍(LmL1mS|1md)uL(p) YLmL(p^) χmS ζ0 . (36)

The last two terms χmS and ζ₀ denote spin and isospin wave functions, respectively. The S and D components of the deuteron wave function (DWF) are given by u₀(P) and u₂(P), respectively. In the present work, we compute the radial deuteron wave functions u_L(p) using the realistic and high-precision Bonn full model ^[⁵⁵^].

For the elementary γN→πN amplitude, we take the π-production operator from the unitary isobar MAID-2007 model introduced in Ref. ^[⁴⁶^]. This model is based on Born terms, ρ and ω vector-meson exchange contributions, and 13 four-star nucleon resonance excitations. It describes well the elementary γN→πN amplitude and agrees with experimental observations very well. The MAID-2007 model is parameterized in terms of invariant amplitudes and therefore allows one to evaluate the transition deuteron amplitude in any frame of reference.

To study the uncertainties caused by the use of different elementary pion photoproduction operators in the analyses of γd→π0d observables, the dynamical DMT-2001 model ^[⁴⁷^] and the χMAID-2013 model ^[⁴⁸^] are used. The DMT-2001 model is a unitary dynamical model based on a non-resonant background described by Born terms and vector-meson exchange contributions in the t-channel (ρ and ω) and the following 8 four-star nucleon resonances in the s-channel: P₃₃(1232), P₁₁(1440), D₁₃(1520), S₁₁(1535), S₃₁(1620), S₁₁(1650), F₁₅(1680) and D₃₃(1700). The DMT-2001 model describes the pion photo- and electroproduction observables in terms of photon and nucleon degrees of freedom and provides a very good description of experimental data in the near-threshold region. The χMAID-2013 model for studying pion photo- and electroproduction on the nucleon in the near-threshold region has been constructed in relativistic chiral perturbation theory and included all multipole amplitudes up to and including l=4 or, in other words, G waves. This model provides a comprehensive description of the elementary process on the free nucleon up to photon energies of 170 MeV in the laboratory frame. The details of each of the used models (MAID-2007, DMT-2001, and χMAID-2013) can be found in their original works in Refs. ^[⁴⁶^-⁴⁸^] and therefore will not repeated here.

3 Results and discussions

In this section, we explore the sensitivity of the results for unpolarized cross sections as well as all various beam, target, and beam-target spin asymmetries of the γd→π0d reaction near threshold to the elementary pion photoproduction amplitude. For this purpose, we use for the elementary reaction amplitudes three different realistic models. These models are the MAID-2007 ^[⁴⁶^], DMT-2001 ^[⁴⁷^], and χMAID-2013 ^[⁴⁸^]. For the deuteron wave function, we use the realistic and high-precision Bonn full model ^[⁵⁵^]. We discuss the energy and angular dependences of the results for these observables and give a comparison of the obtained results with the available experimental data. In all the upcoming figures, the dashed, dotted, and solid curves represent the results using MAID-2007 ^[⁴⁶^], DMT-2001 ^[⁴⁷^], and χMAID-2013 ^[⁴⁸^] models for the elementary operator, respectively.

3.1 Differential and total cross sections

First, we show in Fig. 2 the energy and angular dependences of the results for unpolarized differential cross section, dσ₀/dΩ, using different elementary γN→πN amplitudes. We see that the dσ₀/dΩ results with different elementary amplitudes are quite different especially at forward pion angles. The differences between the results with various elementary amplitudes decrease with increasing pion angle until they become very small at θ = 180^o. When the dashed curve (MAID-2007) is compared with both the dotted (DMT-2001) and solid (χMAID-2013) curves, one can see that these differences are very obvious at forward pion angles and show up the discrepancies among elementary amplitudes. In this case, the calculated dσ₀/dΩ within the MAID-2007 model is smaller than those within DMT-2001 and χMAID-2013. At extreme backward pion angles, we see that the curves represent the results of dσ₀/dΩ using MAID-2007, DMT-2001, and χMAID-2013 are very close to each other and thus the influence of elementary operators on dσ₀/dΩ is negligible in this case. This means that the dσ₀/dΩ results are sensitive to the choice of the elementary amplitude at forward pion angles.

Figure 2 (Color online) The differential cross section for the reaction γd→π0d using different elementary operators and the DWF from Bonn full potential. The dashed, dotted, and solid curves represent the results using the MAID-2007, DMT-2001, and χMAID-2013 models for the elementary operator, respectively. Results at E_γ = 144 MeV are multiplied by the factor in the parentheses.

Figure 3 shows the results for the unpolarized total cross section, σ₀, for the reaction γd→π0d as a function of the photon energy in the laboratory system, E_γ, using different elementary operators. The importance of the choice of elementary amplitude is clearly addressed when the calculation with the MAID-2007 (dashed curve) is compared to the DMT-2001 (dotted curve) and χMAID-2013 (solid curve) curves. It is very clear that the results using MAID-2007 model differ significantly from the DMT-2001 and χMAID-2013 ones. However, the σ₀ results using DMT-2001 and χMAID-2013 models are close to each other. We would like to point out that the MAID-2007 model is based on a single channel approach, whereas the DMT-2001 model is based on coupled channels. This is an important difference between both models.

Figure 3 (Color online) The unpolarized total cross section for the reaction γd→π0d as a function of photon lab-energy using different elementary operators and the DWF from Bonn full NN potential. Curve conventions as in Fig. 2.

3.2 Single-spin asymmetries

Now, we focus our attention on the single-spin asymmetries of polarized photons (Σ) or polarized deuterons (T₁₁, T_20, T₂₁ and T₂₂) for γ→d→π0d and γd→→π0d, respectively. We present in Fig. 4 the results for the linear photon asymmetry Σ at different photon lab-energies as a function of emission pion angle θ in the γd c.m. frame. In general, one can see the results for Σ remain same qualitatively but quantitatively slightly changed. Figure 4 shows that the photon Σ-asymmetry has dominately negative values in the photon energy domain of the present work. One displays also that the Σ-asymmetry decreases with increasing pion scattering angle until a minimum close to θ ≃ 130^o at E_γ = 144 MeV is reached. This minimum is shifted towards lower pion angles with increasing the photon lab-energy. Then the photon asymmetry Σ increases with increasing pion angle until it reaches zero at θ = π. It is also noticeable from Fig. 4 that the photon asymmetry Σ vanishes at θ = 0^o and 180^o, because in that case the differential cross section cannot depend on the azimuthal angle ϕ, since at θ = 0^o and π the angle ϕ is undefined or can be arbitrary. At extreme forward and backward emission pion angles, one notes that the photon Σ-asymmetry is relatively small in comparison to the results when θ varies from 60^o to 150^o.

Figure 4 (Color online) Same as in Fig. 2 but for the photon Σ-asymmetry with linearly polarized photons and unpolarized deuterons.

We found that the asymmetry Σ is sensitive to the elementary amplitude, especially in the peak region where sizeable differences are obtained in the pion angle range from 60^o to 150^o. It has a minimal value around θ ≃ 135^o in the case of χMAID-2013 (solid curve) and it is shifted towards lower pion angles in the case of DMT-2001 (dotted curve) and MAID-2007 (dashed curve). It is also very obvious that the computations with different elementary amplitudes are quite different with, in absolute size, a larger Σ-asymmetry predicted using χMAID than the ones obtained with DMT and MAID models. This discrepancy shows up the differences among elementary pion photoproduction operators and means that Σ is very sensitive to the choice of the elementary amplitude, in particular in the vicinity of the peak.

For polarization observables with polarized deuteron targets and unpolarized photon beams, we present in Fig. 5 the sensitivity of the results for the vector T₁₁ and tensor T_2M (M = 0,1,2) deuteron spin asymmetries for the reaction γd→→π0d to the choice of elementary pion photoproduction amplitude. We see that the results for T₁₁ asymmetry exhibit qualitatively similar behaviors for different elementary pion photoproduction operators. The T₁₁ asymmetry is sensitive to the imaginary parts of the scattering amplitudes and its values vanish identically at θ = 0 and π. This asymmetry depends on the relative phase of the matrix elements as can be seen from Eq. (12). It would vanish for a constant overall phase of the t-matrix. One can also see that the results using different elementary operators are quantitatively rather different even at forward and backward pion angles. This discrepancy displays the differences among elementary pion photoproduction operators which means that the T₁₁-asymmetry is sensitive to the choice of the elementary amplitude.

Figure 5 (Color online) Same as in Fig. 2 but for the deuteron asymmetries with vector T₁₁ and tensor T_2M (M = 0, 1, 2) polarized deuterons and unpolarized photons.

The results for tensor deuteron spin asymmetries T_2M (M = 0,1,2) are also shown in Fig. 4. In general, we see that the results for these asymmetries at the lowest photon energy, E_γ = 144 MeV, are rather different than the results at higher photon energies, especially in the case of T₂₂ asymmetry. The results for T₂₀ and T₂₁ asymmetries using various elementary amplitudes exhibit qualitatively, but not quantitatively, similar behaviors. One can see that the results using different elementary operators are quantitatively rather different, which means that these spin asymmetries are slightly sensitive to the choice of the elementary amplitude. When the photon energy increases, we see that the curves represent the results of T₂₀ and T₂₁ using various elementary amplitudes are close to each other and thus the influence of elementary operators on T₂₀ is small in this case. The T₂₂ asymmetry indicates at photon energies greater than 144 MeV that it has an oscillatory shape. In contrast to the T₂₀ and T₂₁ cases, we find that the results for T₂₂ asymmetry using different elementary amplitudes exhibit qualitatively and quantitatively different behaviors and show up the sensitivity of its results to the elementary amplitude.

A serial system of four columns packed with Eichhornia crassipes and 16 cm radio and 150 cm length is capable of keeping an outlet concentration below 10 ppb

3.3 Beam-target double-spin asymmetries

Here, we report the numerical results for the beam-target double spin asymmetries for γ→d→→π0d reaction near threshold. We start with the results for T10c and T11c asymmetries with circular polarized photons and vector polarized deuterons as shown in Fig. 6 as functions of pion angle in the γd c.m. frame at the same photon lab-energies as mentioned before in the case of single-spin asymmetries. We see that the results for T10c and T11c at different photon lab-energies remain same qualitatively but quantitatively slightly changed. The T10c and T11c asymmetries behave the same as T₂₀ and T₂₁ ones, respectively. Figure 6 displays that the T10c asymmetry has values between —1 and 1. It begins negative at θ = 0^o and increases with increasing pion angle until a maximum value at θ ≃ 90^o. Then, it rapidly falls down to negative values when increasing the pion angles. The maximum value of T10c is shifted towards higher pion angles at E_γ = 144 MeV. As mentioned in the Introduction, the spin asymmetry T10c determines the GDH sum rule ^[⁴⁹^]. The T11c results vanish at θ = 0^o and π and indicate that it has an oscillatory shape. The T10c and T11c results using different elementary operators are quantitatively rather different at the lowest photon energy, which means that these asymmetries are sensitive to the choice of the elementary amplitude close to threshold. By increasing photon energies, the curves represent the results of T10c and T11c using different pion production amplitudes are close to each other and thus a small dependence of these asymmetries on the elementary operators is obtained.

Figure 6 (Color online) Same as in Fig. 2 but for the beam-target double-spin asymmetries with circularly polarized photons and vector polarized deuterons.

Figure 7 shows the results for the beam-target double spin asymmetries T21c and T22c for circularly polarized photons and tensor polarized deuterons. A quick glance reveals that the results for T21c and T22c are negative and vanish at θ = 0^o and π. In contrast to the vector spin asymmetries T10c and T11c, it is obvious from Fig. 7 that the tensor spin asymmetries T21c and T22c are much more sensitive to the choice of elementary amplitude. Figure 7 shows also that the sensitivity of T21c and T22c results to the elementary pion photoproduction amplitude is very important, especially in the pion angle range between 30^o and 120^o.

Figure 7 (Color online) Same as in Fig. 2 but for the beam-target double-spin asymmetries with circularly polarized photons and tensor polarized deuterons. Results for the T22c asymmetry are multiplied by the factor in the parentheses.

In Fig. 8 we present our results for T10l, T1+1l, and T1-1l double spin asymmetries with longitudinal polarized photons and vector polarized deuterons. In general, we see that these double spin asymmetries are sensitive to the elementary amplitude, especially in the pion angle range between 30^o and 150^o. The asymmetry T10l is smaller than the other ones and it differs in size between the results with different elementary amplitudes. This emphasizes the very important dependence of the T10l results on the elementary amplitude. As discussed in Refs. ^[³¹^,³⁸^], the vector asymmetries T1Ml are considerably small for π⁰ production near threshold. Figure 8 displays also that the difference between the results for T10l and T1-1l asymmetries using χMAID-2013 model and both DMT-2001 and MAID-2007 increases with increasing photon lab-energy, which indicates that these spin asymmetries are very sensitive the choice of the elementary γN→πN amplitude.

Figure 8 (Color online) Same as in Fig. 2 but for the beam-target double-spin asymmetries with linearly polarized photons and vector polarized deuterons. Results for the T1-1l asymmetry are multiplied by the factor in the parentheses.

The results for the beam-target double spin asymmetries T20l, T2+1l, T2+2l, T2-1l, and T2-2l with longitudinal polarized photons and tensor polarized deuterons are shown in Fig. 9. We see that these spin asymmetries are also affected by the choice of various elementary amplitudes, especially at photon energies close to threshold. The sensitivity of the results for these asymmetries to the elementary amplitude is very obvious at the lowest photon energy, E_γ = 144 MeV, and decreases with increasing photon energy. This sensitivity is much more obvious in the case of T20l and T2-1l asymmetries since the differences between the results with various elementary operators are significant, in particular in the peak region. The T2±2l asymmetries are sensitive to both the real and the imaginary parts of the scattering amplitudes.

Figure 9 (Color online) Same as in Fig. 2 but for the beam-target double-spin asymmetries with linearly polarized photons and tensor polarized deuterons. Results for the T2-1l and T2-2l asymmetries are multiplied by the factor in the parentheses.

Figure 10 shows the results for the beam-target double-spin asymmetries σ0 T~20, σ0 T~10c and σ0 T~22l of the total cross section as functions of the photon lab-energy. We see that the results for these asymmetries with different elementary amplitudes are quite different. The differences between the results with various elementary amplitudes increase with increasing photon lab-energy. When the dashed curve (MAID-2007) is compared with both the dotted (DMT-2001) and solid (χMAID-2013) curves, one can see that these differences are obvious and show up the discrepancies among elementary amplitudes. The calculated σ0 T~20, σ0 T~10c, and σ0 T~22l within MAID-2007 model is, in absolute size, smaller than those within DMT-2001 and χMAID-2013. At photon energies close to threshold, we see that the curves represent the results of σ0 T~20, σ0 T~10c, and σ0 T~22l using MAID-2007, DMT-2001, and χMAID-2013 models are close to each other and thus the influence of elementary operator on these spin asymmetries is negligible in this case.

Figure 10 (Color online) Same as in Fig. 3 but for the beam-target double-spin asymmetries σ0 T~20, σ0 T~10c and σ0 T~22l of the total cross section.

3.4 Helicity-dependent cross sections and E-asymmetry

Next, we present in Fig. 11 the results for the doubly polarized differential cross sections for parallel (upper part) and antiparallel (middle part) spins of photon and deuteron as functions of pion angle in the c.m. frame at various photon lab-energies using different elementary amplitudes. These polarized differential cross sections are given by

dσPdΩ=dσ0dΩ 1+12T20+32T10c , (37)

dσAdΩ=dσ0dΩ 1+12T20-32T10c . (38)

Figure 11 (Color online) Same as in Fig. 2 but for the doubly polarized differential cross sections for parallel (upper part) and antiparallel (middle part) spins of photon and deuteron and their difference (lower part). Results at E_γ = 144 MeV are multiplied by the factor in the parentheses.

In Fig. 11 we also present an important physical observable which is the difference d(σ^P — σ^A)/dΩ (lower part) that has a direct relation with the GDH sum rule ^[⁴⁹^]. The results displayed show the contribution of each dσ^P/dΩ and dσ^A/dΩ on the difference. Since their values have opposite behavior with increasing θ, we can see that the difference has negative values at small pion angles 0^o < θ < 30^o because of dσ^A/dΩ has the larger contribution in this range of pion angles. Thereafter, for θ ranges between 30^o and 90^o, d(σ^P — σ^A)/dΩ starts to increase and behaves the same as dσ^P/dΩ. For θ = 90^o, the values of d(σ^P — σ^A)/dΩ start to decrease due to dσ^A/dΩ which has larger contribution again.

We would like to mention that the threshold region is dominated by the pion production with spin 1/2, which makes the antiparallel term dσ^A/dΩ larger than the parallel one dσ^P/dΩ. The estimated values of d(σ^P — σ^A)/dΩ close to threshold at E_γ = 144MeV are rather small compared to their values at higher energies under consideration.

As for the dσ^P/dΩ results (upper part in Fig. 11), we observe that the results are sensitive to the choice of elementary amplitude at the peak region since we obtain smaller values using MAID-2007 than using DMT-2001 and χMAID-2013. At extreme forward and backward pion angles, similar results are obtained and the sensitivity of dσ^P/dΩ results to the elementary amplitude is negligible. In the case of dσ^A/dΩ results (middle part in Fig. 11), we see that the influence of dσ^A/dΩ on the elementary amplitude is similar to the case of unpolarized differential cross section. One can see that the differences among dσ^A/dΩ results using different elementary amplitudes are very obvious at forward pion angles and the calculation within the MAID-2007 is smaller than those within DMT-2001 and χMAID-2013 but provides similar results at extreme pion backward angles. This discrepancy shows up the differences among elementary amplitudes. The computations of the differential spin asymmetry with respect to circularly polarized photons and oriented deuterons, d(σ^P — σ^A)/dΩ, (lower part in Fig. 11) shows that the sensitivity to the choice of elementary amplitude is also important in the peak position. This means that the difference d(σ^P — σ^A)/dΩ is also sensitive to the choice of the elementary amplitude.

Figure 12 shows the results for the helicity-dependent total cross sections for the γ→d→→π0d reaction as functions of photon lab-energies using different elementary amplitudes. Results are displayed as follows: (left panel) total cross section σ^P for circularly polarized photons on a deuteron target with spin parallel to the photon spin; (middle panel) σ^A, the same for antiparallel spins of photon and deuteron; (right panel) spin asymmetry σ^P — σ^A. We see that the cross sections σ^P and σ^A as well as the deuteron spin asymmetry σ^P — σ^A present similar behaviors for all elementary amplitudes used in the present work. Compared the calculation with the MAID-2007 (dashed curve) to the DMT-2001 (dotted curve) and χMAID-2013 (solid curve) ones, we clearly address the importance of the choice of elementary amplitude. We see that the results using MAID-2007 model differ from the DMT-2001 and χMAID-2013 ones. We also find that the results for σ^P — σ^A starts out negative due to the E₀₊ multipole, which is dominant in the threshold region. The influence of elementary amplitude is important in σ^P, σ^A, and σ^P — σ^A. We would like to emphasize here that several experiments to measure the deuteron spin asymmetry σ^P — σ^A are presently underway ^[³^,⁸^,⁹^].

Figure 12 (Color online) Same as in Fig. 3 but for the doubly polarized total cross sections for parallel (left panel) and antiparallel (middle panel) spins of photon and deuteron and their difference (right panel).

During the recent years, there is a considerable interest in experiments ^[¹^-⁷^] to measure the double polarization observable E for the γ→d→→π0d reaction. This asymmetry is given by

E(θ)=d(σA-σP)/dΩd(σA+σP)/dΩ=d(σA-σP)/dΩ2dσ0/dΩ . (39)

In Fig. 13 we present the results for E-asymmetry as a function of the emission pion angle θ in the γd c.m.frame at four fixed values of the incident photon lab-energy. We see that the E-asymmetry has qualitatively a similar behavior for all incident photon lab-energies considered. Its maximum equals unity at θ = 0^o and 180^o. The curves begin with unity and decrease as the pion angle increases until a minimum value at θ ≃ 120^o is reached. Then, it increases again to unity. The minimum value is shifted towards lower pion angles with increasing E_γ. The negative values in the E-asymmetry come mainly from higher positive contribution in dσ^P/dΩ. As mentioned in Refs. ^[²²^,³⁴^], the beam-target double polarization E-asymmetry is an excellent observable to test any weakness in the underlying elementary pion photoproduction model. Figure 13 shows that the helicity E-asymmetry is sensitive to the choice of the elementary γN→πN amplitude at incident photon lab-energies close to threshold. When the photon energy increases, we see that the sensitivity of the results for E-asymmetry to the choice of the pion photoproduction operator is rather small.

Figure 13 (Color online) Same as in Fig. 3 but for the double polarization E-asymmetry.

3.5 Comparison with experimental data

We close this section by comparing our results with the available experimental data. In fact, several experiments to measure both single and double polarization observables for coherent π⁰-photoproduction on the deuteron are presently underway. Unfortunately, no data are currently available for these polarization observables in the kinematic region under consideration in the present work and therefore we cannot make a comparison to experimental data for polarization observables. Thus, in the present work we compare the calculated results for the unpolarized differential cross section with the available data.

Figure 14 shows a comparison between our results for the unpolarized differential cross section dσ₀/dΩ at two values of photon lab-energies (E_γ = 151.4 and 171.8 MeV) and the experimental data from TAPS ^[⁴^]. One readily sees, that the estimated results for dσ₀/dΩ using different elementary amplitudes underestimate the experimental data at backward pion angles. The results with the χMAID-2013 model (solid curve) are the nearest one to the experimental data in this case. At forward pion angles and E_γ = 151.4 MeV, an overestimation of the results using χMAID-2013 and DMT-2001 elementary amplitudes is found. But the results for dσ₀/dΩ using MAID-2007 model underestimate the last three data points at high angles. At E_γ =171.8 MeV and for forward pion angles, a good agreement with the experimental data is also obtained. In this case, the shape given by MAID model is consistent with data lacking only in strength. We would like to mention that the elementary operators considered in the present work use some values of resonance photocouplings which especially in case of neutron are not yet well understood, and fitting these values to the data could improve the agreement between theory and experiment.

Figure 14 (Color online) The unpolarized differential cross section for γd→π0d in comparison with the experimental data from TAPS ^[⁴^]. Curve conventions as in Fig. 2.

We would like to point out that even at the PWIA level studied in the present work, there are also uncertainties due to the deuteron wave function, since one is involving large momentum components of the deuteron. Therefore, in Fig. 15 we present results for dσ₀/dΩ using different deuteron wave functions in comparison with the experimental data from TAPS ^[⁴^]. For this purpose, we use the deuteron wave functions of the Bonn full ^[⁵⁵^] (solid curve), CD-Bonn ^[⁵⁶^] (dashed curve), and Paris ^[⁵⁷^] (dotted curve) NN potentials. These potentials are exceedingly employed for numerical estimations of electromagnetic reactions on the deuteron, give a precise characterization of the NN scattering data and phase shifts, and used to characterize the range of the NN interaction.

Figure 15 (Color online) The unpolarized differential cross section for γd→π0d using χMAID-2013 elementary amplitude in comparison with the experimental data from TAPS ^[⁴^]. The dotted, dashed, and solid curves represent the results using the Paris, CD-Bonn, and Bonn-full NN potentials for the deuteron wave function, respectively.

It is clear from Fig. 15 that the results using the deuteron wave function of the Paris potential is smaller than those using CD-Bonn and Bonn full potentials. At E_γ = 151.4 MeV, we note an overestimation of the results using CD-Bonn and Bonn full potentials at θ < 60^o, whereas a good agreement with the experimental data using the Paris potential is obtained in this case. On the contrary, a good agreement between the results using CD-Bonn and Bonn full potentials and the experimental data is obtained at E_γ = 171.8 MeV and forward pion angles. The results using the Paris potential underestimate the experimental data in this case. At backward pion angles, we see that the results using various NN potentials underestimate the experimental data. The origin of the differences obtained using various realistic deuteron wave functions maybe due to the tensor force between two nucleons. It is well-known that a practical measure for the strength of the tensor-force component contained in a nuclear potential is the predicted D-state probability of the deuteron, P_D (see, for example, Refs. ^[⁵⁸^,⁵⁹^]). The P_D values for the NN potentials used in this work are 4.85% for CD-Bonn, 4.25% for Bonn full, and 5.77% for Paris. The difference in these values of P_D is related to the different tensor part of the NN potentials. The dependence of the γd→π0d and ed→e'd' observables on the D-state component of the deuteron wave function was investigated in Refs. ^[²⁰^] and ^[⁶⁰^], respectively. It was found that the D-wave contribution becomes visible at backward scattering angles.

The obtained discrepancies between our estimations for dσ₀/dΩ and the experimental data can be attributed to the neglected contributions from two-body effects in the transition M-matrix. It was found in Refs. ^[¹⁶^,⁶¹^] that the influence of intermediate first-order pion rescattering and the two-loop diagram which includes πN and NN rescattering in the intermediate state are very important at photon energies near threshold. In addition, the meson-exchange current effects were found to be quite significant for coherent π⁺ photoproduction on ³He in Ref. ^[⁶²^].

From the preceding discussions it is apparent that the choice of the elementary amplitude has a visible effect on unpolarized cross sections as well as on various beam, target, and beam-target spin asymmetries in the γd→π0d reaction near threshold. Summarizing, we can say that the MAID-2007 model provides different predictions for all possible observables in γd→π0d than the χMAID-2013 and DMT-2001 models and that these observables provide excellent test object for different elementary operators.

4 Summary and outlook

The main topic of this article was to discuss theoretical uncertainties in the analyses of γd→π0d observables near threshold due to the use of elementary γN→πN amplitudes using a model which is based on time-ordered perturbation theory. As input we have used the realistic MAID-2007 model ^[⁴⁶^] for the elementary amplitude and the high-precision Bonn full NN potential ^[⁵⁵^] for the deuteron wave function. We have presented results for the unpolarized differential and total cross sections as well as for all possible beam, target, and beam-target spin asymmetries of the differential and total cross sections in the photon energy region from near π-threshold to 170 MeV. In particular, we have studied the sensitivity of the calculated results to the choice of elementary pion photoproduction amplitude. For this purpose, we have used in addition to the MAID-2007 model ^[⁴⁶^] the realistic elementary amplitudes from DMT-2001 ^[⁴⁷^] and χMAID-2013 ^[⁴⁸^] models. We have also compared our calculations with the available experimental data.

We have found that the estimations of the uncertainty on the γd→π0d observables show important sensitivity to the modeling of the elementary γN→πN operator. In many cases the deviation among results obtained using different elementary amplitudes is very large. It may be possible that the background contribution from the crossed nucleon pole amplitude plays a significant role in the results presented in this work, because the relative importance of direct and crossed nucleon pole amplitudes grows below the Δ-resonance region. The direct nucleon pole amplitude always interferes constructively with the Δ -resonance one, whereas the crossed nucleon pole amplitude interferes constructively for energies below the Δ -resonance region. The differences found in the predictions of γd→πNN observables ^[³⁴^-³⁹^] using different elementary operators are also seen here in the results for γd→π0d observables. The calculated results for the unpolarized differential cross section using different elementary amplitudes are compared with the experimental data from TAPS ^[⁴^]. A satisfactory agreement between theory and experiment is obtained only at forward pion angles. At backward pion angles, the calculations underestimate the experimental data and a disagreement is obtained.

The results presented above highlight the sensitivity of γd→π0d observables to the choice of pion photoproduction amplitude. The obtained differences in the numerical results for γd→π0d observables with various elementary operators can be attributed to the numerical account for the off-shell effects when the elementary nucleon amplitude is embedded into the deuteron process. In fact, it is difficult to give precise quantitative account for these off-shell effects, because the equivalence existing in the on-shell γN→πN operator is broken in deuteron calculations when one nucleon or both of them are off their mass shells.

In summary, we conclude that the calculated results are of particular interest for the evaluation of the systematic uncertainties caused by the use of different elementary operators in the analyses of γd→π0d measurements. The calculated differential cross section does not described the TAPS data ^[⁴^] at backward angles. This discrepancy between theory and experiment maybe resolved by considering the neglected effects from two-body mechanisms in the transition M-matrix. To study the influence of these reaction mechanisms on the γd→π0d observables, a more realistic treatment including all possible reaction mechanisms will be reported in a forthcoming paper ^[⁶³^]. An independent evaluation in the framework of effective field theory would also be very interesting. On the experimental side, precise measurements of γd→π0d observables can check these predictions and also provide a rigorous test of theoretical models.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under grant number R.G.P.1/1/42. E.M. Darwish would like to thank M.I. Levchuk, V.V. Gauzshtein, and M.N. Nevmerzhitsky for many interesting and helpful discussions.

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Received: March 16, 2022; Accepted: March 31, 2022

^* E-mail: darwish@science.sohag.edu.eg.

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Revista mexicana de física

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Rev. mex. fis. vol.68 n.6 México Nov./Dec. 2022 Epub July 31, 2023

https://doi.org/10.31349/revmexfis.68.061201